Triangle Bisectors5.1 (Part 2)

SWBAT• Construct perpendicular bisectors

and angle bisectors of triangles• Apply properties of perpendicular

bisectors and angle bisectors of triangles

•concurrent•point of concurrency

•circumcenter•circumscribed

• incenter• inscribed

Vocabulary

When three or more lines intersect at one point, the lines are said to be concurrent

The point of concurrency is the point where

the lines intersect

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Since a Triangle has three vertices...

It has three Angle Bisectors.

What did you notice about the Angle Bisectors of the Triangles?

5

Angle bisectors start at a vertexAngle bisectors start at a vertexand and bisect the angle.bisect the angle.

IncenterIncenter

Angle Bisectors

The Incenter is the Point of Concurrency of the Angle Bisectors

The incenter is always inside the triangle.

Any point on an angle bisector is Any point on an angle bisector is equidistant from the sides of the angle equidistant from the sides of the angle

This makes the This makes the IncenterIncenterequidistant from all 3 sidesequidistant from all 3 sides

The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.

Since a Triangle has three sides...

It has three Perpendicular Bisectors.

What did you notice about the Perpendicular Bisectors of the

Triangles?

A Perpendicular bisector of a side formsA Perpendicular bisector of a side forms a a 90° angle90° angle and bisects the sideand bisects the side..

CircumcenterCircumcenter

Perpendicular Bisectors

The Circumcenter is the Point of Concurrency of the Perpendicular Bisectors

The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

Recall:Recall: Any point on the Any point on the perpendicular bisectorperpendicular bisector

of a segment is equidistant from theof a segment is equidistant from theendpoints of the segment.endpoints of the segment.

AA

BB

CC DD

AB is the AB is the perpendicularperpendicularbisector of CDbisector of CD

This makes the This makes the CircumcenterCircumcenterequidistant from the 3 verticesequidistant from the 3 vertices

The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.

Survival TrainingSurvival TrainingYou’re Stranded On A Triangular You’re Stranded On A Triangular

Shaped Island. The Rescue Ship CanShaped Island. The Rescue Ship CanOnly Dock On One Side Of The Island Only Dock On One Side Of The Island

But You Don’t Know Which Side. At But You Don’t Know Which Side. At Which Point Of Concurrency Would Which Point Of Concurrency Would

You Set Up Camp So You Are An You Set Up Camp So You Are An EqualEqual Distance From All 3 Sides?Distance From All 3 Sides?

INCENTERINCENTER

What If The Ship Could OnlyWhat If The Ship Could OnlyDock At One Of The Vertices? Dock At One Of The Vertices? Would You Change The Would You Change The Location Of Your Camp ?Location Of Your Camp ?If So, Where?If So, Where?

YESYES CIRCUMCENTERCIRCUMCENTER